Individual Preferences, Monetary Gambles, and Stock Market
Participation: A Case for Narrow Framing
By NICHOLAS BARBERIS, MING HUANG,
AND
RICHARD H. THALER*
We argue that “narrow framing,” whereby an agent who is offered a new gamble
evaluates that gamble in isolation, may be a more important feature of decisionmaking than previously realized. Our starting point is the evidence that people are
often averse to a small, independent gamble, even when the gamble is actuarially
favorable. We find that a surprisingly wide range of utility functions, including many
nonexpected utility specifications, have trouble explaining this evidence, but that
this difficulty can be overcome by allowing for narrow framing. Our analysis makes
predictions as to what kinds of preferences can most easily address the stock market
participation puzzle. (JEL D81, G11)
Economists, and financial economists in particular, have long been interested in how people
evaluate risk. In this paper, we try to shed new
light on this topic. Specifically, we argue that a
feature known as “narrow framing” may play a
more important role in decision-making under
risk than previously realized. In traditional
models, which define utility over total wealth or
consumption, an agent who is offered a new
gamble evaluates that gamble by merging it
with the other risks she already faces and checking whether the combination is attractive. Narrow framing, by contrast, occurs when an agent
who is offered a new gamble evaluates that
gamble to some extent in isolation, separately
from her other risks.
Our starting point is the evidence that people
are often reluctant to take a small, independent
gamble, even when the gamble is actuarially
favorable. For example, in an experimental setting, Amos Tversky and Daniel Kahneman
(1992) find that approximately half of their subjects turn down a small gamble with two
equiprobable outcomes, even though the potential gain is twice the potential loss.
In this paper, we try to understand what kinds
of preferences can explain such attitudes to risk.
It is already understood that, without a property
known as “first-order” risk aversion, it is hard for
preferences to explain the rejection of a small,
independent, actuarially favorable gamble (Larry
G. Epstein and Stanley E. Zin, 1990; Matthew
Rabin, 2000). Our own contributions are: first, to
show that, surprisingly, even utility functions that
do exhibit first-order risk aversion have difficulty
explaining why such a gamble would be rejected;
and, second, that utility functions that combine
first-order risk aversion with narrow framing have
a much easier time doing so.1
The intuition for these results is straightforward.
* Barberis: Yale School of Management, 135 Prospect
Street, P.O. Box 208200, New Haven, CT 06520 (e-mail:
[email protected]); Huang: Johnson School of
Management and SUFE, Sage Hall, Cornell University,
Ithaca, NY 14853 (e-mail: [email protected]); Thaler:
Graduate School of Business, University of Chicago, 5807
South Woodlawn Avenue, Chicago, IL 60637 (e-mail:
[email protected]). Huang acknowledges financial support from the National Natural Science Foundation of
China under grant 70432002. We thank three anonymous
referees, Douglas Bernheim, John Campbell, Darrell Duffie,
Larry Epstein, John Heaton, Daniel Kahneman, David Laibson, Erzo Luttmer, Terrance Odean, Mark Rubinstein, Bill
Sharpe, Jeffrey Zwiebel, and seminar participants at Duke
University, Emory University, Harvard University, INSEAD,
London Business School, London School of Economics, Massachusetts Institute of Technology, New York University,
Northwestern University, Princeton University, Stanford University, University of California at Berkeley, University of
Chicago, University of Pennsylvania, Yale University, and the
NBER for helpful comments.
1
We give the formal definition of first-order risk aversion in Section I. In informal terms, a utility function
exhibits first-order risk aversion if it is locally risk averse,
unlike most standard preferences, which are locally riskneutral; loss aversion, whereby the utility function is kinked
at the current wealth level, is an example of first-order risk
aversion.
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Suppose that an agent with first-order risk aversion, but who does not engage in narrow framing, is offered a small, independent, actuarially
favorable gamble. Suppose also, as is reasonable, that the agent faces some preexisting risk,
such as labor income risk or house price risk. In
the absence of narrow framing, the agent must
evaluate the new gamble by merging it with her
preexisting risk and checking if the combination
is attractive. It turns out that the combination is
attractive: since the new gamble is independent
of the agent’s other risks, it diversifies those
other risks, and, even though first-order risk
averse, the agent finds this useful. The only
way to make the agent reject the gamble is to
choose an extreme parameterization of her
utility function. However, such a parameterization typically implies, counterfactually, the
rejection of highly attractive gambles with
larger stakes.
A simple way out of this difficulty, then, is to
suppose that, when the agent evaluates the small
gamble, she does not fully merge it with her
preexisting risk but, rather, thinks about it in
isolation, to some extent; in other words, she
frames the gamble narrowly. Using a recently
developed preference specification that allows
for both first-order risk aversion and narrow
framing, we confirm that a utility function with
these features can easily explain the rejection of a
small, independent, actuarially favorable gamble,
while also making sensible predictions about attitudes to large-scale gambles: if the agent’s firstorder risk aversion is focused directly on the small
gamble rather than just on overall wealth risk, she
will be reluctant to take it.
Our analysis of independent monetary gambles has useful implications for financial markets. For example, it is useful for understanding
what kinds of preferences can address the stock
market participation puzzle: the fact that, even
though the stock market has a high mean return
and a low correlation with other household
risks, many households have historically been
reluctant to allocate any money to it (N. Gregory Mankiw and Stephen P. Zeldes, 1991; Michael Haliassos and Carol C. Bertaut, 1995).
It is already understood that, without firstorder risk aversion, it is hard to find a preferencebased explanation of this evidence. Our analysis
of small, independent gambles suggests a more
surprising prediction: even preferences that do exhibit first-order risk aversion will have trouble
SEPTEMBER 2006
explaining nonparticipation. It also suggests that
preferences that combine first-order risk aversion
with narrow framing—in this case, narrow framing of stock market risk—will have an easier time
doing so. We confirm these predictions in a simple
portfolio choice setting.
While the term “narrow framing” was first
used by Kahneman and Daniel Lovallo (1993),
the more general concept of “decision framing”
was introduced much earlier by Tversky and
Kahneman (1981). There are already several
cleverly designed laboratory demonstrations of
narrow framing in the literature.2 This paper
shows that a more basic piece of evidence on
attitudes to risk, not normally associated with
narrow framing, may also need to be thought of
in these terms. Moreover, while existing examples of narrow framing do not always have
obvious counterparts in the everyday risks people face, the simple risks we consider do—not
least in stock market risk—making our results
applicable in a variety of contexts.
The idea that a combination of first-order risk
aversion and narrow framing may be relevant
for understanding aversion to small gambles has
been proposed before, most notably by Rabin
and Thaler (2001). Those authors do not, however, demonstrate the sense in which first-order
risk aversion, on its own, is insufficient. This is
the crucial issue we tackle here.
If narrow framing does, sometimes, play a
role in the way people evaluate risk, it is important that we learn more about its underlying
causes. At the end of the paper, we discuss some
interpretations of narrow framing, including the
possibility that it arises when people make decisions intuitively, rather than through effortful
reasoning (Kahneman, 2003). We argue that the
recent attempts to outline a theory of narrow
framing have made framing-based hypotheses
much more testable than they were before.
I. Attitudes to Monetary Gambles
People are often averse to a small, independent gamble, even when the gamble is actuarially favorable. In an experimental setting,
Tversky and Kahneman (1992) find that ap2
Daniel Read et al. (1999) survey many examples of
narrow framing, including those documented by Tversky
and Kahneman (1981) and Donald A. Redelmeier and Tversky (1992).
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BARBERIS ET AL.: A CASE FOR NARROW FRAMING
proximately half of their subjects turn down a
small gamble with two equiprobable outcomes,
even though the potential gain is twice the potential loss. In a more recent study, Mohammed
Abdellaoui et al. (2005) also find that their
median subject is indifferent to a small gamble
with two equiprobable outcomes only when the
gain is twice the potential loss.3
In this paper, we try to understand what kinds
of preferences can explain an aversion to a
small, independent, actuarially favorable gamble. For concreteness, we consider a small, independent gamble whose potential gain is much
less than twice the potential loss, namely4
G S ⫽ 共550, 1 ⁄ 2 ; ⫺500, 1⁄2 兲,
to be read as “gain $550 with probability 1⁄2 and
lose $500 with probability 1⁄2 ,” and ask: what
kinds of preferences can explain the rejection of
this gamble? We do not insist that a utility
function be able to explain the rejection of GS at
all wealth levels, but rather that it do so over a
range of wealth levels—at wealth levels below
$1 million, say. We take “wealth” to mean total
wealth, including both financial assets and such
nonfinancial assets as human capital.
Is it reasonable to posit that individuals tend
to reject GS even at a wealth level of $1 million,
a wealth level that probably exceeds that of
Tversky and Kahneman’s (1992) median subject? In Barberis et al. (2003), an earlier version
of the current paper, we ask four groups of
people— 68 part-time MBA students, 30 financial advisors, 19 chief investment officers at
large asset management firms, and 34 clients of
a U.S. bank’s private wealth management division (the median wealth in this last group ex3
This evidence is not necessarily inconsistent with riskseeking behavior like casino gambling or the buying of
lottery tickets. Lottery tickets are different from Tversky
and Kahneman’s (1992) gambles, in that they offer a tiny
probability of substantial gain, rather than two equiprobable
outcomes. An individual can be averse to a 50:50 bet
offering a gain against a loss, even if she is risk-seeking
over low-probability gains. Gambling is a special phenomenon, in that people would not accept the terms of trade
offered at a casino if those terms were offered by their bank,
say. It must be that, in the casino setting, people either
misestimate their chance of winning or receive utility from
the gambling activity itself.
4
The “S” subscript in GS stands for Small stakes. We
sometimes use the notation X/Y to refer to a 50:50 bet to win
$X or lose $Y. GS is therefore a “550/500” bet.
1071
ceeds $10 million)—about their attitudes to a
hypothetical GS. In all four groups, the majority
reject GS, and even in the wealthiest group, the
rejection rate is 71 percent. Playing GS for real
money with a second group of MBA students
leads to an even higher rejection rate than
among the first group of MBAs.
We focus on preference-based explanations
for aversion to small gambles because the alternative explanations seem incomplete. One alternative view is that the rejection of gambles like
GS is due to transaction costs that might be
incurred if a liquidity-constrained agent has to
finance a loss by selling illiquid assets (Raj
Chetty, 2004). Such a mechanism does not,
however, explain why wealthy subjects with
substantial liquid assets are often averse to GS.
Nor is suspicion a satisfactory explanation for
the rejection of GS: the fear, for example, that
the experimenter is using a biased coin to determine the outcome. Offering to use a subject’s
coin instead does little to change attitudes to the
gamble.5
Of course, many utility functions can capture an
aversion to GS by simply assuming high risk aversion. To ensure that the assumed risk aversion is
realistic, we insist that the utility functions we
consider also make sensible predictions about attitudes to large-scale gambles by, for example,
predicting acceptance of the large, independent
gamble6
G L ⫽ 共20,000,000, 1 ⁄ 2 ; ⫺10,000, 1⁄2 兲
over a reasonable range of wealth levels—
wealth levels above $100,000, say.
In summary, then, we are interested in understanding what kinds of preference specifications
can predict both:
I. That GS is rejected for wealth levels W ⱕ
$1,000,000; and
II. That GL is accepted for wealth levels W ⱖ
$100,000.
5
It is also unlikely that people turn down GS simply
because they do not want, or feel able, to evaluate it. In
our experience, subjects, when debriefed, typically explain their aversion to G S by saying that “the pleasure of
a $550 gain doesn’t compensate for the pain of a $500
loss.” It therefore appears that they do evaluate the gamble,
but find it unattractive.
6
The “L” subscript in GL stands for Large stakes.
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For certain utility specifications, it can make
a difference whether gambles are “immediate”
or “delayed.” A gamble is immediate if its uncertainty is resolved at once, before any further
consumption decisions are made. A delayed
gamble, on the other hand, might play out as
follows: in the case of GS, the subject is told
that, at some point in the next few months, she
will be contacted and informed either that she
has just won $550 or that she has lost $500, the
two outcomes being equally probable and independent of other risks.
Although certain utility functions can predict
different attitudes to immediate and delayed gambles, people do not appear to treat the two kinds of
bets very differently. Barberis et al. (2003), for
example, record very similar rejection rates for
immediate and delayed versions of GS. We therefore look for preference specifications that can
satisfy conditions I and II both when GS and GL
are immediate, and when they are delayed. Since
a delayed gamble’s uncertainty is resolved only
after today’s consumption is set, it must be analyzed in a multiperiod framework. We therefore
work with intertemporal preferences, not static
ones, throughout the paper.
A. Utility Functions
To structure our discussion, we introduce a
simple taxonomy of intertemporal preferences.
Our list is not exhaustive, but it covers most of the
preference specifications used in economics. In
parentheses, we list the abbreviations that we use
to refer to specific classes of utility functions.
SEPTEMBER 2006
recursive structure in which time t utility, V t ,
is defined through
(1)
V t ⫽ W共C t , ␮ 共Ṽ t ⫹ 1 兩I t 兲兲,
where ␮(Ṽt ⫹ 1兩It) is the certainty equivalent of
the distribution of future utility Ṽt ⫹ 1 conditional on time t information It, and W( 䡠 , 䡠 ) is an
aggregator function which aggregates current
consumption Ct with the certainty equivalent of
future utility to give current utility.
The three kinds of recursive utility on our list
differ in the properties they impose on the certainty equivalent functional ␮⵺. In the first
kind, labelled “R-EU,” ␮⵺ has the expected
utility form, so that
(2)
␮ 共X̃兲 ⫽ h ⫺1 Eh共X̃兲,
for some increasing h⵺. In the “R-SORA”
class, ␮⵺ is non-EU but exhibits “secondorder” risk aversion, which means that it predicts risk neutrality for infinitesimal risks. In
simple terms, utility functions with second-order risk aversion are smooth. Finally, we consider the “R-FORA” class, in which ␮⵺ is
non-EU and exhibits “first-order” risk aversion,
which, as noted earlier, means that it predicts
risk aversion even over infinitesimal bets.7
II. The Limitations of Standard Preferences
[R-EU]:
It is already understood that the first three
kinds of utility functions on our list—the EU,
R-EU, and R-SORA classes— have trouble satisfying condition I (the rejection of GS) while
also satisfying condition II (the acceptance of
GL). Rabin (2000), for example, shows that no
increasing, concave utility function in the EU
class is consistent with both conditions. The
intuition is straightforward. An individual with
EU preferences is locally risk-neutral; since GS
Expected utility preferences are familiar
enough. In an intertemporal setting, nonexpected utility is typically implemented using a
7
The labels “second-order” and “first-order” are due to
Uzi Segal and Avia Spivak (1990). Formally, second-order
risk aversion means that the premium paid to avoid an
actuarially fair gamble k␧˜ is, as k 3 0, proportional to k2.
Under first-order risk aversion, the premium is proportional
to k. While non-EU functions can exhibit either first- or
second-order risk aversion, utility functions in the EU class
can generically exhibit only second-order risk aversion: an
increasing, concave utility function can have a kink only at
a countable number of points.
[EU]:
Expected utility preferences
Nonexpected utility preferences:
Recursive utility with EU
certainty equivalent
[R-SORA]: Recursive utility with non-EU,
second-order risk averse
certainty equivalent
[R-FORA]: Recursive utility with non-EU,
first-order risk averse certainty
equivalent
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BARBERIS ET AL.: A CASE FOR NARROW FRAMING
involves small stakes, she would normally take
it. To get her to reject it, in accordance with
condition I, we need very high local curvature.
In fact, her utility function must have high local
curvature at all wealth levels below $1 million
because condition I requires rejection of GS at
all points in that range. Rabin’s (2000) analysis
shows that “linking” these locally concave
pieces gives a utility function with global risk
aversion so high that the agent rejects even the
very favorable large gamble GL.
While R-EU preferences are nonexpected
utility, the fact that, in this case, the certainty
equivalent functional ␮⵺ is in the expected
utility class means that an agent with these
preferences evaluates risk using an expected
utility function. Rabin’s (2000) argument therefore also applies to R-EU preferences: no utility
function in this class is consistent with both
conditions I and II.
R-SORA preferences can, in principle, satisfy
conditions I and II, but only with difficulty. An
agent with R-SORA preferences is locally riskneutral and will therefore normally be happy to
accept a small, actuarially favorable gamble like
GS. To make her reject it, we need very high local
curvature, which, in turn, requires an extreme parameterization. Such a parameterization, however,
almost always implies high global risk aversion,
thereby making the individual reject attractive,
large-scale gambles like GL.8
The main contribution of this paper is to
show that even the fourth preference class on
our list—the R-FORA class, in which the certainty equivalent functional ␮⵺ is non-EU and
first-order risk averse— has difficulty satisfying
conditions I and II. This is a surprising result
because, at first sight, it appears that R-FORA
preferences can easily be consistent with these
conditions. In fact, as noted by Andrew Ang et
al. (2005) and anticipated even earlier by Epstein and Zin (1990), such preferences have no
trouble satisfying conditions I and II, so long as
8
Those R-SORA preferences that are able to satisfy
conditions I and II typically stumble on the following additional observation: that people tend to reject 1.1y/y and
accept 4y/y for a wide range of values of y. R-SORA
preferences find it hard to explain such “linear” attitudes, as
they need to invoke very strong nonlinearity, or local curvature, to capture the rejection of 1.1y/y for just one value of
y. For more discussion of EU, R-EU, and R-SORA preferences, see Barberis et al. (2003).
1073
the gambles are played out immediately, a caveat that will turn out to be critical.
To see what happens when the gambles are
immediate, consider the following R-FORA
preferences:
(3)
W共C, ␮ 兲 ⫽ 共共1 ⫺ ␤ 兲C ␳ ⫹ ␤␮ ␳ 兲 1/ ␳ ,
0 ⫽ ␳ ⬍ 1,
where ␮⵺ takes a form developed by Faruk Gul
(1991),
(4) ␮ 共Ṽ兲 1 ⫺ ␥ ⫽ E共Ṽ 1 ⫺ ␥ 兲 ⫹ 共 ␭ ⫺ 1兲
⫻ E共共Ṽ 1 ⫺ ␥ ⫺ ␮ 共Ṽ兲 1 ⫺ ␥ 兲1共Ṽ ⬍ ␮ 共Ṽ兲兲兲,
0 ⬍ ␥ ⫽ 1,
␭ ⬎ 1.
These preferences are often referred to as “disappointment aversion” preferences: the agent
incurs disutility if the outcome of Ṽ falls below
the certainty equivalent ␮(Ṽ). The parameter ␭
governs the degree of disutility, in other words,
how sensitive the agent is to losses as opposed
to gains. A ␭ ⬎ 1 puts a kink in the utility
function at the certainty equivalent point,
thereby generating first-order risk aversion.
Epstein and Zin (1989), who give a careful
exposition of recursive utility, propose that, to
evaluate an immediate gamble ṽ at time ␶, an
agent with the recursive preferences in (1) inserts an infinitesimal time step ⌬␶ immediately
before time ␶ consumption C␶ is chosen, and
then applies the recursive calculation over this
time step, checking whether the utility from
taking the gamble,
(5) W共0, ␮ 共Ṽ ␶ ⫹ ⌬ ␶ 兲兲 ⫽ W共0, ␮ 共J共W̃ ␶ ⫹ ⌬ ␶ 兲兲兲
⫽ W共0, ␮ 共J共W ␶ ⫹ ṽ兲兲兲,
where J⵺ is the agent’s value function at time
␶ ⫹ ⌬␶, is greater than the utility from not
taking the gamble,
(6)
W共0, ␮ 共V ␶ ⫹ ⌬ ␶ 兲兲 ⫽ W共0, ␮ 共J共W ␶ ⫹ ⌬ ␶ 兲兲兲
⫽ W共0, ␮ 共J共W ␶ 兲兲兲.
The decision therefore comes down to comparing ␮( J(W␶ ⫹ ṽ)) and ␮( J(W␶)).
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Suppose that, aside from ṽ, the agent’s other
investment opportunities are i.i.d. across periods, and that the outcome of ṽ is independent of,
and does not affect, these other opportunities. It
is then straightforward to show that, for t ⱖ ␶ ⫹
⌬␶, the time t value function in the case of
(3)–(4) takes the form
(7)
J共W t 兲 ⫽ AW t
for some constant A. Equations (5) and (6)
immediately imply that the agent evaluates ṽ
by comparing ␮ (W ␶ ⫹ ṽ) and ␮ (W ␶ ). By
taking ṽ to be G S or G L , we see that the
preferences in (3)–(4) can satisfy conditions
I and II in the case of immediate gambles,
so long as there are values of ␥ and ␭ for
which
(8) 共共W␶ ⫹ 550兲1 ⫺ ␥ ⫹ ␭共W␶ ⫺ 500兲1 ⫺ ␥兲1/共1 ⫺ ␥兲
⬍ 共1 ⫹ ␭ 兲 1/共1 ⫺ ␥ 兲 W ␶
holds for all wealth levels below $1 million, and
(9) 共共W␶ ⫹ 20,000,000兲1 ⫺ ␥
⫹ ␭ 共W ␶ ⫺ 10,000兲 1 ⫺ ␥ 兲 1/共1 ⫺ ␥ 兲
⬎ 共1 ⫹ ␭ 兲 1/共1 ⫺ ␥ 兲 W ␶
holds for all wealth levels above $100,000. A
quick calculation confirms that (␥, ␭) ⫽ (2, 2)
satisfies both (8) and (9). Since ␭ controls sensitivity to losses relative to sensitivity to gains,
we need ␭ ⬎ 1.1 so that the 550/500 bet, with its
1.1 ratio of gain to loss, is rejected.
The intuition for why R-FORA preferences
can explain attitudes to GS and GL when these
gambles are played out immediately is straightforward. In the case of EU, R-EU, and R-SORA
preferences, the difficulty is that the agent is
locally risk-neutral, forcing us to push risk aversion over large gambles up to dramatically high
levels in order to explain the rejection of GS, the
550/500 bet. An agent with R-FORA preferences is, by definition, locally risk averse. Risk
aversion over large gambles does not, therefore,
need to be increased very much to ensure that
GS is rejected.
SEPTEMBER 2006
In the special case where GS and GL are played
out immediately, then, preferences with first-order
risk aversion can satisfy conditions I and II. We
now show that, in the more realistic and general
setting where the gambles are played out with
some delay, even preferences with first-order risk
aversion have a hard time satisfying these conditions. In particular, while they can easily explain
aversion to small, immediate gambles, they
have great difficulty—in a sense that we make
precise— capturing aversion to small, delayed
gambles. This is a serious concern because, as
noted in Section I, people are just as averse to
the 550/500 bet when it is played out with delay
as when it is played out immediately.
Before giving an exact statement of the difficulty with R-FORA preferences, we present an
informal, static example to illustrate the idea.
Consider the following one-period utility function exhibiting first-order risk aversion:
再
x
w共x兲 ⫽ 2x
xⱖ0
for x ⬍ 0 .
Such a utility function can easily explain the
rejection of an immediate 550/500 gamble: an
individual with these preferences would assign
the gamble a value of 550( 1⁄2 ) ⫺ 2(500)( 1⁄2 ) ⫽
⫺225, the negative number signalling that the
gamble should be rejected. But how would this
individual deal with a small, delayed gamble?
In answering this, it is important to recall the
essential difference between an immediate and a
delayed gamble. The difference is that, while
waiting for a delayed gamble’s uncertainty to be
resolved, the agent is also likely to be exposed
to other preexisting risks, such as labor income
risk, house price risk, or risk from financial
investments. This is not true in the case of an
immediate gamble, whose uncertainty, by definition, is resolved at once.
For the R-FORA preferences in (3)–(4), this
distinction can have a big impact on whether a
gamble is accepted. Suppose that the agent is
facing the preexisting risk (30,000, 1⁄2 ;
⫺10,000, 1⁄2 ), to be resolved at the end of the
period, and is wondering whether to take on an
independent, delayed 550/500 gamble, whose
uncertainty will also be resolved at the end of
the period. The correct way for her to think
about this is to merge the new gamble with the
preexisting gamble, and to check whether the
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BARBERIS ET AL.: A CASE FOR NARROW FRAMING
combined gamble offers higher utility. Since the
combined gamble is
(30,550, 1⁄4 ; 29,500, 1⁄4 ;
⫺9,450, 1⁄4 ; ⫺10,500, 1⁄4 ),
the comparison is between
30,000共 1 ⁄ 2 兲 ⫺ 2共10,000兲共 1 ⁄ 2 兲 ⫽ 5,000
and
30,550共 1 ⁄ 4 兲 ⫹ 29,500共 1 ⁄ 4 兲 ⫺ 2共9,450兲共 1 ⁄ 4 兲
⫺ 2共10,500兲共 1 ⁄ 4 兲 ⫽ 5,037.5.
The important point here is that the combined
gamble does offer higher utility. In other words,
the agent would accept the small, delayed gamble, even though she would reject an immediate
gamble with the same stakes. The intuition is
that, since the agent already faces some preexisting risk, adding a small, independent gamble
is a form of diversification, which, even if firstorder risk averse, she can enjoy.
This example suggests that, even if the certainty equivalent functional ␮⵺ exhibits firstorder risk aversion, it may be very difficult to
explain the rejection of gambles like GS, other
than in the special case where uncertainty is
resolved immediately. In Proposition 1 below,
we make the nature of this difficulty precise. In
brief, while an individual with R-FORA utility
acts in a first-order risk-averse manner toward
immediate gambles, the presence of preexisting
risks makes her act in a second-order riskaverse manner toward independent, delayed
gambles.
This immediately reintroduces the difficulty
noted earlier in our discussion of preferences
with second-order risk aversion. Since the agent
is second-order risk averse over delayed gambles, and since the delayed gamble GS is small,
she will normally be keen to accept it. In order
to explain why it is typically rejected, we need
to impose very high local curvature, which, in
turn, requires an extreme parameterization.
Such a parameterization, however, usually also
implies high global risk aversion and therefore
the rejection of large gambles as attractive as
GL. We illustrate this difficulty in Section IIA
with a formal example.
1075
PROPOSITION 1: Consider an individual with
the recursive preferences in (1), where W( 䡠 , 䡠 ) is
strictly increasing and twice differentiable with
respect to both arguments, and where ␮⵺ has the
first-order risk averse form in (4).
Suppose that, at time ␶, the individual is
offered an actuarially favorable gamble k␧˜
which is to pay off between time ␶ and ␶ ⫹ 1,
and whose payoffs do not affect, and are independent of, time ␶ information I␶ and future
economic uncertainty. Suppose also that, prior
to taking the gamble, the distribution of time
␶ ⫹ 1 utility, Ṽ␶ ⫹ 1, does not have finite mass at
its certainty equivalent ␮(Ṽ␶ ⫹ 1).
Then, the individual is second-order risk
averse over the new gamble, and, for sufficiently
small k, accepts it.
PROOF:
See the Appendix.
In simple language, the proposition says that
an individual with R-FORA utility is secondorder risk averse over a delayed, independent
gamble, so long as she faces preexisting risk, an
assumption captured here by the statement that
“Ṽ␶ ⫹ 1 does not have finite mass at its certainty
equivalent ␮(Ṽ␶ ⫹ 1).” While Proposition 1 is
proven for just one implementation of firstorder risk aversion, the argument used in the
proof can also be applied, with minor adjustments, to other implementations of first-order
risk aversion. For example, by strengthening the
assumption that “Ṽ␶ ⫹ 1 does not have finite mass
at its certainty equivalent ␮(Ṽ␶ ⫹ 1)” to “Ṽ␶ ⫹ 1
does not have finite mass at any point,” Proposition 1 can be applied when ␮⵺ takes Menahem E. Yaari’s (1987) rank-dependent expected
utility form, which also exhibits first-order risk
aversion.9
An important step in the proof is an assumption as to how an agent with the preferences in
(1) evaluates a delayed gamble ṽ. In their exposition of recursive utility, Epstein and Zin
(1989) do not suggest a specific method. We
therefore adopt the most natural one, which is
9
David A. Chapman and Valery Polkovnichenko (2006)
note that, if Ṽ␶ ⫹ 1 does have finite mass at some point, then
rank-dependent expected utility can satisfy conditions I and
II. For an agent who owns stock or a house, however, the
assumption that Ṽ␶ ⫹ 1 does not have finite mass at any point
would seem to be a reasonable one.
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THE AMERICAN ECONOMIC REVIEW
that the agent merges the delayed gamble with
the other risks she is already taking and checks
whether the combination offers higher utility. In
other words, she applies the recursive calculation over the time step between ␶ and ␶ ⫹ 1, and
then compares the utility from not taking the
gamble,
(10) W共C*␶ , ␮共Ṽ␶ ⫹ 1 兲兲 ⫽ W共C*␶ , ␮共J共W̃␶ ⫹ 1 兲兲兲
⫽ W共C *␶ , ␮ 共J共共W ␶ ⫺ C *␶ 兲R̃ *␶ ⫹ 1 兲兲兲,
where J⵺ is the time ␶ ⫹ 1 value function,
R̃*␶ ⫹ 1 is the return on invested wealth between ␶
and ␶ ⫹ 1, and where asterisks denote optimal
consumption and portfolio choices, to the utility
from taking it,
SEPTEMBER 2006
A. An Example
We now illustrate the difficulty faced by RFORA preferences with a formal, intertemporal
example. In particular, we show that it is hard
for such preferences to explain both the rejection of the delayed gamble GS and the acceptance of the delayed gamble GL; in other words,
to satisfy both conditions I and II at the same
time.
In our example, we again consider an agent
with the R-FORA preferences in (1), (3), and
(4). We assume that, initially, the only investment opportunity available to the agent is a
risky asset with gross return R̃ between t and
t ⫹ 1, where R̃ has the log-normal distribution
(12)
log R̃ ⬃ N共0.04, 0.03兲, i.i.d. over time.
(11) W共C␶†, ␮共Ṽ␶ ⫹ 1 兲兲 ⫽ W共C␶†, ␮共J共W̃␶ ⫹ 1 兲兲兲
⫽ W共C ␶† , ␮ 共J共共W ␶ ⫺ C ␶† 兲R̃ ␶† ⫹ 1 ⫹ ṽ兲兲兲,
where optimal consumption and portfolio
choices are now denoted by dagger signs. We
contrast C*␶ and R̃*␶ ⫹ 1 with C␶† and R̃␶†⫹1 as a
reminder that, if the agent takes the gamble, her
optimal consumption and portfolio choices will
be different from what they are when she does
not take the gamble.10
In this case, the agent’s time t value function
takes the form
(13)
J共W t 兲 ⫽ AW t
for all t, where the constant A is given by
(14)
AW t ⫽ W共C *t , ␮ 共Ṽ t ⫹ 1 兲兲
⫽ W共C *t , ␮ 共J共W̃ t ⫹ 1 兲兲兲
⫽ W共C *t , ␮ 共J共共W t ⫺ C *t 兲R̃兲兲兲
10
Strictly speaking, an agent with the preferences in (1),
(3), and (4) does not have to evaluate the delayed version of
GS by merging it with her preexisting risk. Since the gamble’s uncertainty is resolved at a single instant in the future,
she could insert an infinitesimal time interval around that
future moment of resolution. Since GS would be her only
source of wealth risk over that interval, her first-order risk
aversion would lead her to reject it, consistent with condition I. It is easy, however, to construct a slightly different
gamble that is immune to such manipulations. Consider a
gamble that, at some point in the future, will deliver a win
of $550 or a loss of $500 with equal probability. Suppose
also that, from now until the final payout, the probability of
eventually winning the $550 is continuously reported. If her
preexisting risk also evolves continuously over time, the
agent must necessarily evaluate this 550/500 gamble by
merging it with her preexisting risk, and will therefore
accept it. We have found, however, that subjects are as
averse to this continuously resolved version of GS as to the
immediate and delayed versions. Once again, the preferences in (3)–(4) have trouble satisfying condition I.
⫽ W共C *t , A ␮ 共共W t ⫺ C *t 兲R̃兲兲.
Now suppose that, at time ␶, the agent is
offered a delayed gamble ṽ, where ṽ is either GS
or GL. As in Proposition 1, ṽ pays off between
␶ and ␶ ⫹ 1, and its payoff does not affect, and
is independent of, It and future economic uncertainty. As a result, whether the agent accepts ṽ
or not, her time t value function continues to
take the form in equation (13) for all t ⱖ ␶ ⫹ 1.
From (10), the agent’s utility if she does not
take ṽ is therefore AW␶; from (11), her utility if
she does take it is
(15) W共C ␶† , ␮ 共J共共W ␶ ⫺ C ␶† 兲R̃ ⫹ ṽ兲兲兲
⫽ W共C ␶† , A ␮ 共共W ␶ ⫺ C ␶† 兲R̃ ⫹ ṽ兲兲.
VOL. 96 NO. 4
BARBERIS ET AL.: A CASE FOR NARROW FRAMING
FIGURE 1. ATTITUDES
TO
MONETARY GAMBLES FOR AN AGENT
RISK-AVERSE PREFERENCES
WITH
1077
FIRST-ORDER
Notes: The “⫹” signs mark the parameter values for which an agent with a recursive
utility function with first-order risk-averse certainty equivalent rejects a 50:50 bet to win
$550 or lose $500 at a wealth level of $1 million. The “x” signs mark the parameter values
for which the agent accepts a 50:50 bet to win $20 million or lose $10,000 at a wealth
level of $100,000.
Figure 1 presents the results from comparing
AW␶ and (15) when ṽ is GS or GL. Our computations set ␤ ⫽ 0.9 and ␳ ⫽ ⫺1, but the results
depend little on these choices. The methodology
behind the calculations is described in full in the
Appendix.
The area marked with “⫹” signs shows the
values of ␥ and ␭ for which the agent rejects
the delayed 550/500 gamble. Extreme values
are required to explain this rejection, with ␥
exceeding 150 across a wide range of values
of ␭. The intuition is that, in the presence of
the preexisting risk in (12), the agent acts in a
second-order risk-averse manner toward delayed gambles. In order to explain the rejection of a small, delayed gamble, we therefore
need very high local curvature, which, in turn,
requires extreme parameters.
The area marked with “x” signs shows the
values of ␥ and ␭ for which the agent accepts
the delayed 20 million/10,000 gamble. There is
no overlap between the two shaded regions. In
other words, the parameters needed to satisfy
condition I are so extreme as to also predict very
high global risk aversion, and therefore the rejection of large-stakes gambles as attractive
as GL.
III. Narrow Framing: A Potential Solution
Many utility specifications have difficulty explaining the rejection of a small, independent,
actuarially favorable gamble. What kinds of
preferences can do so? Clearly, first-order risk
aversion is a helpful ingredient: it explains why
small gambles like GS are rejected when played
out immediately. However, it is not enough:
when a first-order risk-averse agent evaluates a
small, delayed gamble, she merges it with her
preexisting risk and, since the resulting diversification is attractive, she happily accepts it. A
simple way of explaining the rejection of such a
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THE AMERICAN ECONOMIC REVIEW
delayed gamble, then, is to suppose that the
agent does not fully merge it with her preexisting risk, but that, to some extent, she evaluates
it in isolation. By “evaluates it in isolation,” we
mean that the agent derives utility directly from
the outcome of the gamble, and not just indirectly via its contribution to total wealth, as in
traditional models. Equivalently, her utility
function depends on the outcome of the gamble
over and above what that outcome implies for
total wealth risk. This, is “narrow framing.”
We now check that preferences that allow for
both first-order risk aversion and narrow framing can easily satisfy conditions I and II,
whether the gambles are played out immediately or with delay. A preference specification
that incorporates both of these features has recently been developed by Barberis and Huang
(2004). In their formulation, time t utility is
given by
冉
冉冘
(16) Vt ⫽ W Ct , ␮(Ṽt ⫹ 1) ⫹ b0 Et
冊冊
v៮ (G̃i,t ⫹ 1) ,
i
where
(17) W共C, y兲 ⫽ 共共1 ⫺ ␤ 兲C ␳ ⫹ ␤ y ␳ 兲 1/ ␳ ,
0 ⫽ ␳ ⬍ 1,
0 ⬍ ␥ ⫽ 1,
再
for
xⱖ0
x⬍0,
isolation from other risks. First-order risk aversion is also introduced, this time through the
piecewise linearity of v៮ ⵺. Since v៮ ⵺ exhibits
first-order risk aversion, there is no need for
␮⵺ to do so; here, ␮⵺ takes a simple power
utility form.
Barberis and Huang (2004) propose that an
agent with these preferences evaluate an immediate gamble x̃ at time ␶ by, as before, inserting
an infinitesimal time interval ⌬␶ and applying
the recursive calculation over this time step. If
the gamble is framed narrowly, the utility from
taking it is
(20) W共0, ␮ 共Ṽ ␶ ⫹ ⌬ ␶ 兲 ⫹ b 0 E ␶ 共v៮ 共x̃兲兲兲
⫽ W共0, ␮ 共J共W̃ ␶ ⫹ ⌬ ␶ 兲兲 ⫹ b 0 E ␶ 共v៮ 共x̃兲兲兲
⫽ W共0,␮共J共W␶ ⫹ x̃兲兲 ⫹ b0 E␶ 共v៮ 共x̃兲兲兲,
where J⵺ is the time ␶ ⫹ ⌬␶ value function,
while the utility from not taking it is
(21) W共0, ␮ 共V ␶ ⫹ ⌬ ␶ 兲兲 ⫽ W共0, ␮ 共J共W ␶ 兲兲兲.
A delayed gamble at time ␶ is evaluated, as
before, by applying the recursive calculation
over the time step from ␶ to ␶ ⫹ 1. If the gamble
is framed narrowly, the utility from taking it is
(22) W共C ␶† , ␮ 共Ṽ ␶ ⫹ 1 兲 ⫹ b 0 E ␶ 共v៮ 共x̃兲兲兲
(18) ␮ 共Ṽ兲 ⫽ 共E共Ṽ 1 ⫺ ␥ 兲兲 1/共1 ⫺ ␥ 兲 ,
x
(19) v៮ 共x兲 ⫽ ␭x
SEPTEMBER 2006
␭ ⬎ 1,
and where G̃i,t ⫹ 1 are specific gambles faced by
the agent whose uncertainty will be resolved
between time t and t ⫹ 1.
The term prefixed by b0 in (16) shows that,
relative to the usual recursive specification in
(1), utility can now depend on outcomes of
gambles G̃i,t ⫹ 1 over and above what those outcomes mean for total wealth risk: G̃i,t ⫹ 1 now
enters the utility function directly, and not just
indirectly via time t ⫹ 1 utility, Ṽt ⫹ 1. In other
words, the specification in (16) allows for narrow framing, with the parameter b0 controlling
the degree of narrow framing: a b0 of 0 means
no narrow framing at all, while a large b0 means
that G̃i,t ⫹ 1 is evaluated almost completely in
⫽ W共C ␶† , ␮ 共J共W̃ ␶ ⫹ 1 兲兲 ⫹ b 0 E ␶ 共v៮ 共x̃兲兲兲
⫽ W共C␶†,␮共J共共W␶ ⫺ C␶†兲R̃␶† ⫹ 1 ⫹ x̃兲兲
⫹ b 0 E ␶ 共v៮ 共x̃兲兲兲,
where J⵺ is the time ␶ ⫹ 1 value function, R̃␶†⫹1
is the return on invested wealth between ␶ and
␶ ⫹ 1, and where dagger signs denote optimal
consumption and portfolio choices; and the utility from not taking it is
(23) W共C *␶ , ␮ 共Ṽ ␶ ⫹ 1 兲兲
⫽ W共C *␶ , ␮ 共J共共W ␶ ⫺ C *␶ 兲R̃ *␶ ⫹ 1 兲兲兲,
where optimal consumption and portfolio
choices are now denoted by asterisks. We contrast C*␶ and R̃*␶ ⫹ 1 with C␶† and R̃␶†⫹1 as a reminder that optimal consumption and portfolio
choices change when the gamble is accepted.
VOL. 96 NO. 4
BARBERIS ET AL.: A CASE FOR NARROW FRAMING
Using these expressions, we can check that
the preferences in (16)–(19) can indeed satisfy
conditions I and II, whether GS and GL are
immediate or delayed, so long as the gambles
are framed narrowly.11 To see the intuition,
suppose that the 550/500 bet, whether immediate or delayed, is framed narrowly, so that the
agent thinks about it in isolation, to some extent.
Since v៮ ⵺ is steeper for losses than for gains, the
potential loss of $500 looms larger than the
potential gain of $550, leading the agent to
reject the gamble. In other words, if the agent’s
first-order risk aversion is focused directly on
the 550/500 bet rather than just on her overall
wealth risk, she will be reluctant to take the bet.
We consider the same environment as in the
example of Section IIA. We assume that, initially, the agent’s only investment opportunity
is a risky asset with gross return R̃ between t and
t ⫹ 1, where R̃ has the distribution
(24)
log R̃ ⬃ N共0.04, 0.03兲, i.i.d. over time.
Barberis and Huang (2004) show that, in this
case, the agent’s time t value function takes the
form
(25)
J共W t 兲 ⫽ AW t
for all t, where the constant A is given by
(26)
AW t ⫽ W共C *t , ␮ 共Ṽ t ⫹ 1 兲兲
⫽ W共C *t , ␮ 共J共W̃ t ⫹ 1 兲兲兲
⫽ W共C *t , ␮ 共J共共W t ⫺ C *t 兲R̃兲兲兲
⫽ W共C *t ,A ␮ 共共W t ⫺ C *t 兲R̃兲兲.
For simplicity, we assume here that the agent
does not frame her preexisting risk narrowly.
Now suppose that, at time ␶, the agent is
offered a gamble ṽ, where ṽ is either GS or GL
and is either immediate or delayed. As before,
the outcome of ṽ does not affect, and is independent of, It and future economic uncertainty.
As a result, whether the agent accepts ṽ or not,
her time t value function continues to be given
11
To satisfy conditions I and II, we only need the agent
to frame GS narrowly. For the sake of treating the two
gambles symmetrically, we suppose that the agent frames
both of them narrowly.
1079
by equation (25) for t ⱖ ␶ ⫹ ⌬␶ in the case of
an immediate gamble, and for t ⱖ ␶ ⫹ 1 in the
case of a delayed gamble. Expressions (20)–
(23) then allow us to determine the agent’s
attitudes to immediate and delayed versions of
GS and GL. The methodological details are described in full in the Appendix.
We set ␤ and ␳, which have little direct
influence on attitudes to risk, to 0.9 and ⫺1,
respectively. Our calculations then show that,
for a range of values of b0—including a b0 as
low as 0.001—it is easy to find parameter pairs
(␥, ␭) that satisfy conditions I and II, whether
the gambles are immediate or delayed. For example, when b0 ⫽ 0.1, the pair (␥, ␭) ⫽ (1.5, 3)
can do so. The intuition is that a ␭ of 3 generates
enough sensitivity to losses to reject the 550/
500 bet, with its 1.1 gain-to-loss ratio, when that
bet is evaluated to some extent in isolation; but
it does not generate nearly enough sensitivity to
losses to reject the highly attractive 20 million/
10,000 gamble.
IV. Application: Stock Market Participation
We have argued that a wide range of utility
functions, including even those with first-order
risk aversion, have difficulty explaining an
aversion to a small, independent, actuarially
favorable gamble while also making sensible
predictions about attitudes to large gambles.
Utility functions that exhibit both first-order
risk aversion and narrow framing offer a simple
way out of this difficulty.
We now show that our analysis has useful
implications for financial markets, for example,
for the stock market participation puzzle; the fact
that, even though the stock market has a high mean
return, many households have historically been reluctant to allocate any money to it (Mankiw and
Zeldes, 1991; Haliassos and Bertaut, 1995).12
12
Mankiw and Zeldes (1991) report that, in 1984, only
28 percent of households held any stock at all, and only 12
percent held more than $10,000 in stock. Nonparticipation
was not simply the result of not having any liquid assets:
even among households with more than $100,000 in liquid
assets, only 48 percent held stock. Today, the fraction of
households that own stock is closer to 50 percent. The
participation puzzle is therefore primarily the puzzle of
why, historically, many people did not invest in equities,
but also of why, today, people participate more. In this
section, we focus on the first part of the puzzle, and return
to the second part in Section V.
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THE AMERICAN ECONOMIC REVIEW
One approach to the participation puzzle invokes transaction costs of investing in the
stock market; another examines whether nonstockholders have background risk that is correlated with the stock market (John C. Heaton
and Deborah J. Lucas, 1997, 2000; Annette
Vissing-Jorgensen, 2002). These approaches
almost certainly explain some of the observed
nonparticipation, but they may not be able to
account for all of it. Polkovnichenko (2005)
finds that, even among wealthy households, for
whom transaction costs are low, there is still
substantial nonparticipation; and Stephanie
Curcuru et al. (2005) question whether the correlation of the stock market with the background risk of nonstockholders is high enough
to explain an equity allocation as low as zero.
Here, we investigate a third approach to the
participation puzzle: a preference-based approach. In particular, we try to understand what
kinds of preferences can generate nonparticipation in the stock market for parameterizations
that we would consider reasonable. We take a
reasonable parameterization to be one that
makes sensible predictions about attitudes to
large gambles; for example, one that satisfies
condition II, acceptance of the 20 million/
10,000 gamble at wealth levels above $100,000.
It is already understood that preferences that
exhibit only second-order risk aversion have
difficulty addressing the participation puzzle
(Heaton and Lucas, 1997, 2000). Our earlier
analysis suggests a more surprising prediction:
that even preferences with first-order risk aversion will have trouble addressing this puzzle. It
also suggests that preferences that combine
first-order risk aversion with narrow framing—in this case, narrow framing of stock market risk—will have an easier time doing so.
To see the logic behind these predictions,
note that, in the absence of narrow framing, an
agent must evaluate stock market risk by merging it with her preexisting risk and checking if
the combination is attractive. For most households, stock market risk has a correlation close
to zero with other important risks, such as labor
income risk, proprietary income risk, and house
price risk (Heaton and Lucas, 2000). A small
equity position is therefore diversifying, and,
according to our earlier analysis, the agent will
find this attractive, even if first-order risk
averse. Only an extreme parameterization of
SEPTEMBER 2006
her preferences will make her withdraw from
the stock market entirely. Such a parameterization, however, almost always implies high
global risk aversion, thereby violating condition II.
A simple way out of this difficulty is to
suppose that, when the agent evaluates the stock
market, she does not fully merge its risk with
the other risks she is already facing but, rather,
thinks about it in isolation, to some extent. By
focusing the agent’s first-order risk aversion
directly on the stock market rather than just on
her overall wealth risk, we can generate nonparticipation more easily.
In making these predictions, we are assuming
that our analysis of independent gambles is also
relevant for gambles that are merely relatively
uncorrelated with other risks. While this is
likely to be true, the only way to be sure is to
check our predictions explicitly in a simple
portfolio choice setting. This is what we now
do.
Consider an agent who, at the start of each
period, has a fixed fraction ␪៮ N of her wealth tied
up in a nonfinancial asset— our so-called preexisting risk—with gross return
(27)
log R̃N,t ⫹ 1 ⫽ gN ⫹ ␴N␧˜ N,t ⫹ 1 ,
and who is wondering what fraction ␪S of her
wealth to invest in the stock market, which has
gross return
(28)
log R̃S,t ⫹ 1 ⫽ gS ⫹ ␴S␧˜ S,t ⫹ 1 ,
where
(29)
冉 ␧␧˜˜ 冊 ⬃ N冉冉 00冊 , 冉 ␻1 ␻1 冊冊 ,
N,t
S,t
i.i.d. over time.
The remaining fraction of her wealth, 1 ⫺ ␪៮ N ⫺ ␪S,
is invested in a risk-free asset earning a gross
return of Rf , so that the return on total wealth
between t and t ⫹ 1 is
(30)
R̃ W,t ⫹ 1 ⫽ 共1 ⫺ ␪៮ N ⫺ ␪ S 兲R f
⫹ ␪៮ N R̃ N,t ⫹ 1 ⫹ ␪ S R̃ S,t ⫹ 1 .
VOL. 96 NO. 4
BARBERIS ET AL.: A CASE FOR NARROW FRAMING
In reality, of course, the fraction of an individual’s wealth made up by a nonfinancial asset
like a house is likely to vary over time. Fixing it
at ␪៮ N is a simplifying assumption, but is not
crucial for our results.
We solve this portfolio problem for three
different preference specifications: (a) as a
benchmark, the power utility form
冘
⬁
(31)
E0
t⫽0
␤t
C 1t ⫺ ␥
;
1⫺␥
(b) recursive utility with first-order risk averse
certainty equivalent, or R-FORA, as in (1), (3),
and (4); and (c), preferences that combine firstorder risk aversion with narrow framing of
stock market risk, which, following the formulation in (16), can be written as
(32)
Vt ⫽ W共Ct , ␮共Ṽt ⫹ 1 兲 ⫹ b0 Et 共v៮ 共G̃S,t ⫹ 1 兲兲兲,
where G̃S,t ⫹ 1, the stock market gamble the
agent is taking, is given by
(33)
G̃S,t ⫹ 1 ⫽ ␪S 共Wt ⫺ Ct 兲共R̃S,t ⫹ 1 ⫺ Rf 兲,
and where W( 䡠 , 䡠 ), ␮⵺, and v៮ ⵺ are given in
(17)–(19).13
To match historical annual data, we set the
mean gS and volatility ␴S of log stock market
returns to 0.06 and 0.2, respectively. We set the
mean gN and volatility ␴N of log returns on the
nonfinancial asset to 0.04 and 0.03, respectively, and the fraction of nonfinancial wealth in
total wealth, ␪៮ N, to 0.75, but our results depend
little on the values of these three parameters. A
more important parameter is ␻, the correlation
of stock market risk with the agent’s preexisting
risk. Heaton and Lucas (2000) report correlations between stock market risk and three
13
The simplest way to define the stock market gamble is
␪S(Wt ⫺ Ct)(R̃S,t ⫹ 1 ⫺ 1): the capital allocated to the stock
market multiplied by the net return on the stock market. For
tractability, we adopt the slight modification of defining the
gain or loss on the stock market gamble relative to the
risk-free rate Rf. The interpretation is that a stock market
return is only considered a gain, and only delivers positive
utility, if it is higher than the return on T-Bills.
1081
major kinds of preexisting risk—labor income, proprietary income, and real estate— of
⫺0.07, 0.14, and ⫺0.2 in the aggregate, respectively. They also find that, in the cross
section of households, these correlations
rarely exceed 0.2, and, in their simulations,
consider only correlations in the range [⫺0.1,
0.2]. An ␻ of 0.1 is therefore a reasonable
benchmark value. Finally, we set the gross
risk-free rate R f to 1.02.
For these return process parameters, and for
each utility function in turn, we compute the
range of preference parameters for which the
agent allocates a fraction ␪S ⱕ 0 of overall
wealth to the stock market, in other words, the
range of parameters for which, even though the
stock market offers a high mean rate of return,
gS ⫽ 0.06, she still refuses to participate in it.
(The solution technique used for preference
specifications (b) and (c) is described in the
Appendix.) We then check whether these preference parameters are reasonable, in other
words, whether they are consistent with condition II. Specifically, we check whether an agent
with wealth of $100,000, a fraction ␪៮ N of which
is invested in the nonfinancial asset with return
distribution in (27), and a fraction 1 ⫺ ␪៮ N of
which is invested in the risk-free asset, would
accept a 20 million/10,000 gamble. As in Sections IIA and III, we use ␤ ⫽ 0.9, ␳ ⫽ ⫺1, and
b0 ⫽ 0.1.
Power utility preferences illustrate the basic
puzzle. For the return process parameters above,
␥ ⱖ 137 is required for a ␪S ⱕ 0 allocation to
the stock market; but for such ␥, the agent is
very risk averse, and would turn down a 20
million/10,000 gamble at a wealth level of
$100,000, violating condition II.
Figures 2 and 3 present results for preferences (b) and (c), respectively. In each figure,
the “⫹” signs indicate the parameters for which
the agent chooses a ␪S ⱕ 0 allocation to the
stock market, while the “x” signs show the
parameters for which she accepts GL.
Figure 2 confirms that, for R-FORA preferences, with first-order risk aversion but no
narrow framing, it is hard to generate nonparticipation for reasonable parameter values. In
fact, for this implementation of first-order risk
aversion, it is impossible: there is no overlap
between the two shaded regions. In the absence of narrow framing, a positive position
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FIGURE 2. ATTITUDE
TO
STOCK MARKET RISK FOR AN AGENT
RISK-AVERSE PREFERENCES
SEPTEMBER 2006
WITH
FIRST-ORDER
Notes: The “⫹” signs mark the parameter values for which an agent with a recursive utility
function with first-order risk-averse certainty equivalent does not participate in a stock market
offering a high mean return and low correlation with other risks. The “x” signs mark the
parameter values for which the agent accepts a 50:50 bet to win $20 million or lose $10,000
at a wealth level of $100,000.
in the stock market is diversifying and therefore desirable, even for a first-order riskaverse agent. Extreme parameter values are
needed to keep the agent out of the stock
market.
The result in Figure 2 may be especially
surprising, given that some authors, including
Ang et al. (2005), appear to show that Gul
(1991)-type first-order risk aversion, the type
we work with here, does address the stock market participation puzzle. In fact, there is no
inconsistency. Earlier research has focused on
the special case where the agent has no preexisting risk, but simply chooses between riskless
T-Bills and a risky stock market. In this case,
first-order risk aversion over total wealth risk
implies first-order risk aversion over stock market risk, and can therefore indeed generate stock
market nonparticipation. Figure 2 shows that, in
the more realistic case where the agent has
preexisting risk, this result breaks down: first-
order risk aversion can no longer produce nonparticipation for reasonable parameters.14
Figure 3 confirms our second prediction: that,
as soon as narrow framing of the stock market is
allowed, a wide range of parameter values can
14
Chapman and Polkovnichenko (2006) argue that another implementation of first-order risk aversion, Yaari’s
(1987) rank-dependent expected utility, can generate nonparticipation more easily. This is only true, however, if the
distribution of the agent’s future utility, Ṽ␶ ⫹ 1, has finite
mass at some point. If Ṽ␶ ⫹ 1 does not have finite mass at any
point, rank-dependent expected utility encounters the same
difficulty as Gul’s (1991) disappointment aversion in explaining nonparticipation (Haliassos and Christis Hassapis,
2001). For an agent who owns a house or a private business,
the assumption that Ṽ␶ ⫹ 1 does not have finite mass at any
point would seem to be a reasonable one. An approach that
can generate nonparticipation even for continuously distributed preexisting risk is one based on ambiguity aversion
(Epstein and Martin Schneider, 2006). This approach works
in much the same way as our own, by inducing first-order
risk aversion over the stock market gamble itself.
VOL. 96 NO. 4
BARBERIS ET AL.: A CASE FOR NARROW FRAMING
FIGURE 3. ATTITUDE
TO
STOCK MARKET RISK FOR
FRAMING PREFERENCES
AN
AGENT
WITH
1083
NARROW
Notes: The “⫹” signs mark the parameter values for which an agent with a recursive utility
function that allows for both first-order risk aversion and narrow framing does not participate
in a stock market offering a high mean return and low correlation with other risks. The “x”
signs mark the parameter values for which the agent accepts a 50:50 bet to win $20 million
or lose $10,000 at a wealth level of $100,000.
deliver a ␪S ⱕ 0 equity allocation, while still
predicting acceptance of GL, the 20 million/
10,000 gamble. For example, the parameter triple
(␥, ␭, b0) ⫽ (1.5, 3, 0.1), which satisfies not only
condition II but also condition I, predicts nonparticipation. The intuition is that a ␭ of 3 generates
enough sensitivity to losses to make the stock
market appear unattractive when evaluated to
some extent in isolation, but not nearly enough to
reject GL.
We have focused here on the issue of stock
market participation. The analysis in Sections II
and III, however, applies in the same way to any
situation where people are reluctant to take on a
small, actuarially favorable gamble that is only
weakly correlated with preexisting risks: the tendency, for example, of some stockholders to hold
only a small number of stocks, rather than the
many stocks recommended for diversification; the
refusal of many stock-owning households to diversify their holdings with a position in interna-
tional equities; or the surprisingly low deductibles
many consumers choose in their automobile insurance policies (Alma Cohen and Liran Einav,
2005). In each of these cases, first-order risk aversion, on its own, cannot easily explain the facts;
but first-order risk aversion combined with narrow
framing— of individual stocks, of foreign equity
holdings, or of the financial risk of a car accident— can do so more readily.
Barberis and Huang (2006) point out that, in
many representative agent models of the stock
market, the equity premium is determined by
the agent’s attitude, in equilibrium, to taking
on an extra dollar of stock market risk that is
only weakly correlated with her existing holdings. If these models are to match the high
historical equity premium, the agent must be
strongly averse to this small amount of weakly
correlated risk. The analysis in Section II suggests that first-order risk aversion, alone, is unlikely to generate the aversion needed for a high
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premium, a prediction confirmed by Epstein and
Zin (2001) and Barberis and Huang (2006). The
analysis in Section III, however, suggests that a
combination of first-order risk aversion and narrow framing— here, narrow framing of stock
market risk—will more easily generate the required aversion, and hence also, a high premium. Shlomo Benartzi and Thaler (1995) and
Barberis et al. (2001) confirm that models based
on loss aversion over narrowly framed stock
market risk can indeed generate sizeable equity
premia.
V. Interpreting Narrow Framing
We have argued that preferences that combine first-order risk aversion with narrow framing may be helpful for understanding attitudes
to both independent monetary gambles and the
stock market. Of the two features, narrow framing is the more unusual. We therefore end by
discussing its interpretation in more detail.
One way narrow framing can arise is if an
agent takes nonconsumption utility, such as regret, into account. Regret is the pain we feel
when we realize that we would be better off
today if we had taken a different action in the
past. Even if a gamble that the agent accepts is
just one of many risks that she faces, it is still
linked to a specific decision, namely the decision to accept the gamble. As a result, it exposes
the agent to possible future regret: if the gamble
turns out badly, the agent may regret the decision to accept it. Consideration of nonconsumption utility therefore leads quite naturally to
preferences that depend on the outcomes of
gambles over and above what those outcomes
mean for total wealth.
Another view of narrow framing is proposed
by Kahneman (2003). He suggests that it arises
when decisions are made intuitively, rather than
through effortful reasoning. Since intuitive
thoughts are by nature spontaneous, they are
heavily shaped by the features of the situation at
hand that come to mind most easily—to use the
technical term, by the features that are most
“accessible.” When an agent is offered a 50:50
bet to win $550 or lose $500, the outcomes of
the gamble, $550 and $500, are instantly accessible; much less accessible, however, is the distribution of future outcomes the agent faces
after integrating the 550/500 bet with all of her
other risks. As a result, if the agent thinks about
SEPTEMBER 2006
the gamble intuitively, the distribution of the
gamble, taken alone, may play a more important
role in decision-making than would be predicted by traditional utility functions defined
only over wealth or consumption.
By providing the outline of a theory of framing, Kahneman (2003) makes framing a more
testable concept than it was before. Thaler et al.
(1997) illustrate the kind of test one can do. In
an experimental setting, they ask subjects how
they would allocate between a risk-free asset
and a risky asset over a long time horizon, such
as 30 years. The key manipulation is that some
subjects are shown draws from the distribution
of short-term asset returns—the distribution of
monthly returns, say—while others are shown
draws from a long-term return distribution—the
distribution of 30-year returns, say. Since they
have the same decision problem, the two groups
of subjects should make similar allocation decisions: subjects who see short-term returns
should simply use them to infer the more relevant long-term returns. If this requires too much
effort, however, Kahneman’s (2003) framework
suggests that these subjects will instead use the
returns that are most accessible to them, namely
the short-term returns they were shown. Since
losses occur more often in high-frequency data,
they will perceive the risky asset to be especially risky and will allocate less to it. This is
exactly Thaler et al.’s (1997) finding.
In Section IV, we addressed the stock market
participation puzzle by saying that agents may
get utility from the outcome of their stock market investments over and above what that outcome means for their overall wealth; in other
words, they may frame the stock market narrowly. Is this a plausible hypothesis?
It seems to us that both the “regret” and
“accessibility” interpretations of narrow framing could indeed apply to decisions about the
stock market. Allocating some fraction of her
wealth to the stock market constitutes a concrete
action on the part of the agent— one that she
may later regret if her stock market gamble
turns out poorly.15
15
Of course, investing in T-Bills may also lead to regret
if the stock market goes up in the meantime. Regret is
thought to be stronger, however, when it stems from having
taken an action—for example, moving one’s savings from
the default option of a riskless bank account to the stock
market—than from not having taken an action—for exam-
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BARBERIS ET AL.: A CASE FOR NARROW FRAMING
Alternatively, given our daily exposure, from
newspapers, books, and other media, to information about the distribution of stock market
risk, such information is very accessible. Much
less accessible is any information as to the distribution of future outcomes once stock market
risk is merged with the other kinds of risk that
people often face. Applying Kahneman’s (2003)
framework, judgments about how much to invest
in the stock market might therefore be made, to
some extent, using a narrow frame. Over time, of
course, people may learn that their intuitive decision-making is leading them astray, and may
switch to the normatively superior strategy of participating in the stock market. This may be one
factor behind the rise in stock market participation
over the past 15 years.
The so-called “disposition effect”—the tendency of individual investors to sell stocks in
their portfolios that have risen in value since
purchase, rather than fallen—suggests that
people may not only frame overall stock market risk narrowly, but individual stock risk
as well: perhaps the simplest way of explaining the disposition effect is to posit that
people receive direct utility from realizing a
gain or loss on an individual stock that
they own.16
Alok Kumar and Sonya S. Lim (2005) illustrate another approach to testing framingbased theories. They point out that, under the
hypothesis that the disposition effect is due to
narrow framing, people who are more susceptible to narrow framing will display the dis-
ple, leaving one’s savings in place at the bank. In short,
errors of commission are more painful than errors of omission (Kahneman and Tversky, 1982).
16
Terrance Odean (1998) argues that other potential
explanations of the disposition effect— explanations based
on informed trading, taxes, rebalancing, or transaction
costs—fail to capture important features of the data.
1085
position effect more. They identify these
more susceptible traders as those who tend to
execute just one trade, as opposed to several
trades, on any given day: if a trader executes
just one trade on a given day, the gain or loss
for that trade will be more accessible, making
narrow framing more likely. The data confirm
the prediction: the “one-trade-a-day” traders
exhibit the disposition effect more.
VI. Conclusion
We argue that narrow framing, whereby an
agent who is offered a new gamble evaluates
that gamble in isolation, separately from other
risks she already faces, may be a more important feature of decision-making than previously
realized. Our starting point is the evidence that
people are often averse to a small, independent
gamble, even when the gamble is actuarially
favorable. We find that a surprisingly wide
range of utility functions, including many nonexpected utility specifications, have trouble explaining this evidence; but that this difficulty
can be overcome by allowing for narrow framing. Our analysis makes predictions as to what
kinds of preferences can most easily address the
stock market participation puzzle, as well as
other related financial puzzles. We confirm
these predictions in a simple portfolio choice
setting.
Our analysis does not prove that narrow framing is at work in the case of monetary gambles,
nor that it is at work in the case of stock market
nonparticipation. Given the difficulties faced by
standard preferences, however, the narrow framing view may need to be taken more seriously
than before. With the emergence of new theories
of framing, such as that of Kahneman (2003), we
expect to see new tests of framing-based hypotheses. Such tests should, in time, help us learn more
about the role that narrow framing plays in individual decision-making.
APPENDIX
PROOF OF PROPOSITION 1:
We prove the proposition for a certainty equivalent functional ␮⵺ more general than (4), namely
(34)
u共␮共Ṽ兲兲 ⫽ E共u共Ṽ兲兲 ⫹ 共␭ ⫺ 1兲 E共共u共Ṽ兲 ⫺ u共␮共Ṽ兲兲兲1共Ṽ ⬍ ␮共Ṽ兲兲兲, ␭ ⬎ 1,
where u⵺ has a positive first derivative and a negative second derivative. For
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THE AMERICAN ECONOMIC REVIEW
u共x兲 ⫽
(35)
x1 ⫺ ␥
,
1⫺␥
SEPTEMBER 2006
0 ⬍ ␥ ⫽ 1,
this reduces to (4).
Note first that, since Ṽ␶ ⫹1 does not have finite mass at ␮(Ṽ␶ ⫹1), a small change in the period ␶ ⫹ 1
value function, ⌬Ṽ␶ ⫹ 1 ⫽ ⌬Ṽ(W̃␶ ⫹ 1, I␶ ⫹ 1), changes the certainty equivalent by
(36)
⌬␮ ⫽
E共u⬘共Ṽ␶ ⫹ 1 兲⌬Ṽ␶ ⫹ 1 兲 ⫹ 共␭ ⫺ 1兲 E共u⬘共Ṽ␶ ⫹ 1 兲1共Ṽ␶ ⫹ 1 ⬍ ␮兲⌬Ṽ␶ ⫹ 1 兲
⫹ o共储⌬Ṽ␶ ⫹ 1储兲,
u⬘共␮兲共1 ⫹ 共␭ ⫺ 1兲Prob共Ṽ␶ ⫹ 1 ⬍ ␮兲兲
where ␮ denotes ␮(Ṽ␶ ⫹ 1), 储 x储 ⫽ E(兩x兩), and limx30(o( x)/x) ⫽ 0, by definition.
Denote the gamble k␧˜ described in the proposition by ṽ. Assume, for now, that the agent does not
optimally adjust her time ␶ consumption and portfolio choice if she takes the gamble. Then,
⌬Ṽ ␶ ⫹ 1 ⫽ J W 共W̃ ␶ ⫹ 1 , I ␶ ⫹ 1 兲ṽ ⫹ o共储ṽ储兲,
(37)
which, from (36), implies
(38)
⌬␮ ⫽
E共u⬘共Ṽ␶ ⫹ 1 兲JW 共W̃␶ ⫹ 1 , I␶ ⫹ 1 兲ṽ兲 ⫹ 共␭ ⫺ 1兲 E共u⬘共Ṽ␶ ⫹ 1 兲1共Ṽ␶ ⫹ 1 ⬍ ␮兲JW 共W̃␶ ⫹ 1 , I␶ ⫹ 1 兲ṽ兲
⫹ o共储ṽ储兲.
u⬘共␮兲共1 ⫹ 共␭ ⫺ 1兲Prob共Ṽ␶ ⫹ 1 ⬍ ␮兲兲
Since ṽ is independent of other economic uncertainty, we have
(39)
⌬␮ ⫽ E共ṽ兲
E共u⬘共Ṽ␶ ⫹ 1 兲JW 共W̃␶ ⫹ 1 , I␶ ⫹ 1 兲兲 ⫹ 共␭ ⫺ 1兲E共u⬘共Ṽ␶ ⫹ 1 兲1共Ṽ␶ ⫹ 1 ⬍ ␮兲JW 共W̃␶ ⫹ 1 , I␶ ⫹ 1 兲兲
⫹ o共储ṽ储兲,
u⬘共␮兲共1 ⫹ 共␭ ⫺ 1兲Prob共Ṽ␶ ⫹ 1 ⬍ ␮兲兲
so that, to first order, the certainty equivalent of Ṽ␶ ⫹ 1 depends only on the expected value of the
gamble ṽ, not on its standard deviation.
To complete the proof, note that the aggregator function W( 䡠 , 䡠 ) does not generate any first-order
dependence on the standard deviation of ṽ. In addition, assuming that the agent adjusts her time ␶
consumption and portfolio choice optimally when accepting ṽ introduces only terms of the second
order of ṽ. The agent is therefore second-order risk averse over ṽ.
COMPUTING ATTITUDES
TO
MONETARY GAMBLES
Recursive Utility with First-Order Risk Averse Certainty Equivalent
We now describe the methodology behind the delayed gamble calculations of Section IIA. An
important parameter is the constant A in (13). From (14), A satisfies
(40)
AW t ⫽ max W共Ct , A␮共共Wt ⫺ Ct 兲R̃兲兲 ⫽ max兵共1 ⫺ ␤兲C ␳t ⫹ ␤共Wt ⫺ Ct 兲␳A␳共␮共R̃兲兲 ␳其1/␳
Ct
Ct
⫽ maxWt兵共1 ⫺ ␤兲␣ ␳ ⫹ ␤共1 ⫺ ␣兲␳A␳共␮共R̃兲兲␳其1/␳,
␣
where ␣ is the fraction of wealth consumed at time t. The first-order condition is
(41)
共1 ⫺ ␤兲␣␳ ⫺ 1 ⫽ ␤共1 ⫺ ␣兲␳ ⫺ 1A␳␮␳,
where ␮ ⫽ ␮(R̃). When substituted into (40), this gives
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BARBERIS ET AL.: A CASE FOR NARROW FRAMING
1087
A ⫽ 共1 ⫺ ␤ 兲 1/ ␳ ␣ 共 ␳ ⫺ 1兲/ ␳ .
(42)
Substituting this into (41) leads to
␣ ⫽ 1 ⫺ ␤ 1/共1 ⫺ ␳ 兲 ␮ ␳ /共1 ⫺ ␳ 兲 .
(43)
Therefore, given ␮, which can be computed from its definition in (4), we obtain ␣ from (43) and A
from (42).
If the agent does not take the delayed gamble ṽ, her utility at time ␶ is AW␶. If she does take ṽ,
her utility, from (15), is
(44)
max W共C␶ , A␮共共W␶ ⫺ C␶ 兲R̃ ⫹ ṽ兲兲
C␶
冉冉
再
⫽ maxW␶ (1 ⫺ ␤)␣ ␳ ⫹ ␤(1 ⫺ ␣) ␳A␳ ␮ R̃ ⫹
␣
冊冊 冎
ṽ
W␶(1 ⫺ ␣)
␳ 1/␳
⫽ ÂW␶ ,
where A was computed above. This maximization can be performed numerically. We can then
compare  to A to see if the agent should take the delayed gamble.
Recursive Utility with Both First-Order Risk Aversion and Narrow Framing
We now describe the methodology behind the gamble calculations of Section III. An important
parameter is the constant A in (25). From (26), A satisfies
(45) AW t ⫽ max W共Ct , A␮共共Wt ⫺ Ct 兲R̃兲兲 ⫽ max 兵共1 ⫺ ␤兲C␳t ⫹ ␤共Wt ⫺ Ct 兲␳A␳E共R̃1⫺␥兲␳/共1⫺␥兲其1/␳
Ct
Ct
⫽ max Wt兵共1 ⫺ ␤兲␣␳ ⫹ ␤共1 ⫺ ␣兲␳A␳E共R̃1⫺␥兲␳/共1⫺␥兲其1/␳,
␣
where ␣ is the fraction of wealth consumed at time t. The first-order condition is
(46)
共1 ⫺ ␤ 兲 ␣ ␳ ⫺ 1 ⫽ ␤ 共1 ⫺ ␣ 兲 ␳ ⫺ 1 A ␳ 共E共R̃ 1 ⫺ ␥ 兲兲 ␳ /共1 ⫺ ␥ 兲 ,
which, when substituted into (45), gives
A ⫽ 共1 ⫺ ␤ 兲 1/ ␳ ␣ 共 ␳ ⫺ 1兲/ ␳ .
(47)
Substituting this into (46) leads to
(48)
␣ ⫽ 1 ⫺ ␤1/共1 ⫺ ␳兲共E共R̃1 ⫺ ␥兲兲␳/共共1 ⫺ ␥兲共1 ⫺ ␳兲兲.
Therefore, we obtain ␣ from (48) and A from (47).
From (20) and (21), the agent takes an immediate gamble x̃ if and only if
(49)
A关E共W ␶ ⫹ x̃兲 1 ⫺ ␥ 兴 1/共1 ⫺ ␥ 兲 ⫹ b 0 E ␶ 共v៮ 共x̃兲兲 ⬎ AW ␶ ,
where A was computed above.
We now turn to the case of delayed gambles. From (23), if the agent does not take a delayed
gamble x̃, her utility is AW␶, where A was computed above. If she takes the delayed gamble, her
utility, from (22), is
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THE AMERICAN ECONOMIC REVIEW
SEPTEMBER 2006
max W共C␶ , A␮共共W␶ ⫺ C␶ 兲R̃ ⫹ x̃兲 ⫹ b0 E␶ 共v៮ 共x̃兲兲兲
(50)
C␶
冋冉 冉
再
冉冉
⫽ max W␶ (1 ⫺ ␤)␣␳ ⫹ ␤(1 ⫺ ␣)␳ A E R̃ ⫹
␣
⫹ b0 E␶ v៮
冊冊册 冎
冊冊
1⫺␥
x̃
W␶(1 ⫺ ␣)
1/(1⫺␥)
␳ 1/␳
x̃
W␶(1 ⫺ ␣)
⫽ ÂW␶ ,
where A was computed above. This maximization can be performed numerically. We can then
compare  to A to see if the agent should take the delayed gamble.
PORTFOLIO CHOICE CALCULATIONS
We now describe the methodology behind the portfolio choice calculations of Section IV.
Recursive Utility with First-Order Risk Averse Certainty Equivalent
Epstein and Zin (1989) show that, in the i.i.d. setting of Section IV, the consumption-wealth ratio
is a constant ␣, the fraction of wealth allocated to the stock market is a constant ␪S, and the time t
value function is J(Wt) ⫽ AWt for all t. The agent’s problem becomes
(51)
max W共Ct , ␮共Ṽt ⫹ 1 兲兲 ⫽ max W共Ct , A␮共W̃t ⫹ 1 兲兲
Ct ,␪S
Ct ,␪S
⫽ max Wt 关共1 ⫺ ␤兲␣␳ ⫹ ␤共1 ⫺ ␣兲␳A␳共␮共R̃W,t ⫹ 1 兲兲␳兴1/␳,
␣,␪S
where R̃W,t ⫹ 1 is defined in (30). The consumption and portfolio problems are therefore separable.
The portfolio problem is
max ␮共R̃W,t ⫹ 1 兲,
(52)
␪S
which, given the definition of ␮⵺ in (4), can be solved in a straightforward fashion.
Recursive Utility with Both First-Order Risk Aversion and Narrow Framing
Barberis and Huang (2004) show that, in the i.i.d. setting of Section IV, the consumption-wealth
ratio is a constant ␣, the fraction of wealth allocated to the stock market is a constant ␪S, and the time
t value function is J(Wt) ⫽ AWt for all t. The agent’s problem becomes
(53)
AW t ⫽ max W共Ct , ␮共Ṽt ⫹ 1 兲 ⫹ b0 Et 共v៮ 共G̃S,t ⫹ 1 兲兲兲 ⫽ max W共Ct , A␮共W̃t⫹1兲 ⫹ b0 Et 共v៮ 共G̃S,t⫹1兲兲兲
Ct ,␪S
Ct ,␪S
⫽ max Wt 关共1 ⫺ ␤兲␣␳ ⫹ ␤共1 ⫺ ␣兲␳共B*兲␳兴1/␳,
␣
where G̃S,t ⫹ 1 is defined in (33) and
(54)
1⫺␥
1/共1 ⫺ ␥兲
⫹ b0␪S Et 共v៮ 共R̃S,t ⫹ 1 ⫺ Rf 兲兲.
B* ⫽ max A关E共R̃W,t
⫹ 1兲兴
␪S
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1089
The only difficulty with the portfolio problem in (54) is that it depends on the value function
constant A. To handle this, note that the first-order condition for consumption in (53) is
共1 ⫺ ␤兲␣␳ ⫺ 1 ⫽ ␤共1 ⫺ ␣兲␳ ⫺ 1共B*兲␳.
(55)
Substituting this into (53) gives
A ⫽ 共1 ⫺ ␤ 兲 1/ ␳ ␣ 共 ␳ ⫺ 1兲/ ␳ .
(56)
The problem can now be solved as follows. Guess a candidate value of ␣, substitute it into (56) to
generate a candidate A, and then solve portfolio problem (54) for that A. Take the B* that results and
use equation (55) to generate a new ␣. Continue this iteration until convergence occurs. The
converged values represent an optimum.
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